Is Einstein Still Right?

by Clifford M. Will

  The day-to-day operation of the program was done by Hulse, while Taylor made periodic trips down from Amherst throughout the summer to see how things were going. On 2 July, Hulse was by himself when the instruments recorded a very weak pulsed signal. If the signal had been more than 4 percent weaker, it would have fallen below the automatic cutoff that had been built into the search routine and would not even have been recorded. Despite its weakness, it was interesting because it had a surprisingly short period, only 0.059 seconds. At the time, only the Crab pulsar had a shorter period. This made it worth a second look, but it was 25 August before Hulse got around to it.

  The goal of the 25 August observing session was to try to refine the period of the pulses. If this were a pulsar, its period should be the same to at least six decimal places, or to better than a microsecond, over several days, because even if it were slowing down as quickly as the Crab was, the result would be a change only in the seventh decimal place. That is where the troubles began. Between the beginning and the end of the two-hour observing run, the computer analyzing the data produced two different periods for the pulses, differing by almost 30 microseconds. Two days later, Hulse tried again, with even worse results. As a result, he had to keep going back to the original discovery page in his lab notebook and cross out and re-enter new values for the period. His reaction was natural: annoyance. Because the signal was so weak, the pulses were not clean and sharp like those from other pulsars, and the computer must have had problems getting a fix on the pulses. Perhaps this source was not worth the hassle. If Hulse had actually adopted this attitude and dumped the candidate, he and Taylor would have been the astronomical goats of the decade. As it turned out, the suspicious Hulse decided to take an even closer look.

  During the next several days, Hulse wrote a special computer program designed to get around any problems that the standard program might be having in resolving the pulses. But even with the new program, data taken on 1 and 2 September also showed a steady decrease of about 5 microseconds during the two-hour runs. This was much smaller than before, but still larger than it should be, and it was a decrease instead of the expected increase. To continue to blame this on the instruments or the computer was tempting, but not very satisfying.

  But then Hulse spotted something. There was a pattern in the changes of the pulse period! The sequence of decreasing pulse periods on 2 September appeared to be almost a repetition of the sequence of 1 September, except it occurred 45 minutes earlier. Hulse was now convinced that the period change was real and not an artifact. But what was it? Had he discovered some new class of object: a manic depressive or bipolar pulsar with periodic highs and lows? Or was there a more natural explanation for this bizarre behavior?

  The fact that the periods nearly repeated themselves gave Hulse a clue to an explanation. The source was indeed a well-adjusted pulsar, but it wasn’t alone! The pulsar, Hulse postulated, was in orbit about a companion object, and the variation in the observed pulse period was simply a consequence of the Doppler shift (see Figure 5.1). When the pulsar is approaching us, the observed pulse period is decreased (the pulses are jammed together a bit), and when it is receding from us the pulse period is increased (the pulses are stretched out a bit). Actually, optical astronomers are very familiar with this phenomenon in ordinary stars. As many as half the stars in our galaxy are in binary systems (systems with two stars in orbit about each other), and because it is rarely possible to resolve the two stars telescopically, they are identified by the up-and-down Doppler shifts in the frequencies of the spectral lines of the atoms in the atmospheres of the stars. In most ordinary stellar binary systems, the Doppler shifts of the spectra of both stars are observed; however, occasionally one of the stars is too faint to be seen, so astronomers can detect the motion of only one of the stars. And in recent years many exoplanets have been deduced by observing only the Doppler shifted spectra of the parent star. Such appeared to be the case here, where the pulse period was playing the same role as the spectral line in an ordinary star. One of Hulse’s problems with this hypothesis was a practical one: he couldn’t find any decent books on optical stellar binary systems in the Arecibo library because radio astronomers don’t usually concern themselves with such things.

  Figure 5.1 Orbit of a binary system such as the one containing the binary pulsar. The orbit of each body is an ellipse, and their velocities are shown here with arrows. The center of mass C of the system is the focus of each ellipse, while the periastron of one body is denoted P, and the apastron is A.

  Now, because the dish of the Arecibo telescope is built into a natural bowl-shaped valley in the mountains of Puerto Rico, it could only look at the source when it was within 1 hour on either side of the zenith or overhead direction (hence the two-hour runs), and so Hulse couldn’t just track the source for hours on end; he could only observe it during the same two-hour period each day. But the shifting of the sequence of periods in the 1 and 2 September data meant that the orbital period of the system must not be commensurate with 24 hours, and so each day he could examine a different part of the orbit, if indeed his postulate was right. On Thursday 12 September he began a series of observations that he hoped would unravel the mystery (see Figure 5.2).

  Figure 5.2 Pulse period changes of the binary pulsar over a five-day period in September 1974. Adapted from a page of Russell Hulse’s notebook.

  On 12 September the pulse period stayed almost constant during the entire run. On 14 September, the period started from the previous value and decreased by 20 microseconds over the 2 hours. The next day, 15 September, the period started out a little lower and dropped 60 microseconds, and near the end of the run it was falling at the rate of 1 microsecond per minute. The speed of the pulsar along the line of sight must be varying, first slowly, then rapidly. The binary hypothesis was looking better and better, but Hulse wanted to wait for the smoking gun, the clinching piece of evidence. So far the periods had only decreased. But if the pulsar is in orbit, its motion must repeat itself, and therefore he would eventually be able to see a phase of the orbit when the pulse period increased, ultimately returning to its starting value to continue the cycle.

  He didn’t have long to wait. The very next day, 16 September, the period dropped rapidly by 70 microseconds, and with only about 25 minutes left in the observing run it suddenly stopped decreasing, and within 20 minutes it had climbed back up by 25 microseconds. This was all Hulse needed, and he called Taylor in Amherst to break the news. Taylor flew immediately down to Arecibo, and together they tried to complete the solution of this mystery. However, the real excitement was still to come.

  The first thing they determined was the orbital period, by finding the shortest interval over which the pattern of pulse readings repeated. The answer was 7.75 hours, so the 45-minute daily shift that Hulse had seen was just the difference between three complete orbits and one Earth day.

  The next obvious step was to track the pulse period variations throughout the orbit to try to determine the velocity of the pulsar as a function of time. This is a standard approach in the study of ordinary binary systems, and a great deal of information can be obtained from it. If we adopt Newtonian gravitation theory for a moment, then we know that the orbit of the pulsar about the center of mass of the binary system (a point somewhere between the two, depending upon their relative masses) is an ellipse with the center of mass at one focus (see Figure 5.1). The orbit of the companion is also an ellipse about this point, but because the companion is unseen, we don’t need to consider its orbit directly. The orbit of the pulsar lies in a plane that can have any orientation in the sky. It could lie on the plane of the sky, in other words perpendicular to our line of sight, or we could be looking at the orbit edge on, or its orientation could be somewhere between these extremes. We can eliminate the first case, because if it were true, then the pulsar would never approach us or recede from us and we would not detect any Doppler shifts of its period. We can also forget the second case, because if it were true, the
n at some point the companion would pass in front of the pulsar (an eclipse) and we would lose its signal for a moment. No such loss of the signal was seen anywhere during the eight-hour orbit. So the orbit must be tilted at some angle relative to the plane of the sky.

  That is not all that can be learned from the behavior of the pulsar period. Remember that the Doppler shift tells us only the component of the pulsar velocity along our line of sight; it is unaffected by the component of the velocity transverse to our line of sight. Suppose for the sake of argument that the orbit were a pure circle. Then the observed sequence of Doppler shifts would go something like this: starting when the pulsar is moving transverse to the line of sight, we see no shift; one-quarter period later it is moving away from us, and we see a negative shift in the period; one-quarter cycle after that it is again moving transverse, and we see no shift; one-quarter cycle later it is moving toward us with the same velocity, so there is an equal positive shift in the period; after a complete orbital period of seven and three-quarter hours, the pattern repeats itself. The pattern of Doppler shifts in this case is a nice symmetrical one, and totally unlike the actual pattern observed.

  The observed pattern tells us that the orbit is actually highly elliptical or eccentric. In an elliptical orbit, the pulsar does not move on a fixed circle at a constant distance from the companion; instead it approaches the companion to a minimum separation at a point called periastron (the analogue of perihelion for the planets) and separates from the companion one-half of an orbit later to a maximum distance at a point called apastron. At periastron, the velocity of the pulsar increases to a maximum, and following periastron it decreases again, all over a short period of time, while at apastron, the velocity slowly decreases to a minimum value and afterward it slowly increases again.

  The observed behavior of the Doppler shift with time indicated a large eccentricity (see Figure 5.3). Over a very short period of time (only two hours out of the eight) the Doppler shift went quickly from zero to a large value and back, while over the remaining six hours, it changed slowly from zero to a smaller value in the opposite sense and back. In fact, the 16 September smoking gun observation saw the pulsar pass through periastron, while the 12 September observations saw the pulsar moving slowly through apastron. Detailed study of this curve showed that the separation between the two bodies at apastron was four times larger than their separation at periastron. It also showed that the direction of the periastron was almost perpendicular to the line of sight, because the periastron (the point of most rapid variation in velocity) coincided with the largest Doppler shift (the point where the pulsar has the smallest amount of transverse motion).

  Figure 5.3 Location of the pulsar in its orbit. On 12 September, the pulsar is moving through apastron; its speed is low and slowly varying, so there is little change in the observed period (see Figure 5.2). The pulsar is moving away from us, so the period is longer than the “rest” period. On 14 September the pulsar is moving almost transversely, so there is little Doppler shift, and the period is shorter than before. On 15 September, the pulsar is starting to move toward us and is speeding up as it nears periastron; the pulse period decreases markedly toward the end of the run. On 16 September the pulsar starts out moving almost transversely, then quickly passes through periastron, so its velocity toward us quickly reaches a maximum, then decreases; the pulse period rapidly reaches a minimum and then increases. The portion of the orbit seen during the same two-hour interval each day varies because the orbital period is 7.75 hours, so the portion seen is 45 minutes further advanced each day.

  At this point, things began to get very interesting. The actual value of the velocity with which the pulsar was approaching us, as inferred from the decrease in its pulse period, was about 300 kilometers per second, or about one-thousandth of the speed of light! The velocity of recession was about 75 kilometers per second. These are high velocities! The speed of the Earth in its orbit about the Sun is only 30 kilometers per second. Combining the speed information with the orbital period, one could estimate that the average separation between the pulsar and its companion was only about as large as the radius of the Sun.

  When news of this discovery began to spread in late September 1974 it caused a sensation, especially among general relativists. The reason is that the high orbital speeds and close proximity between the two bodies made this a system where effects of general relativity could be measurable.

  In fact, even before Hulse and Taylor’s discovery paper on the binary pulsar appeared in print (but too late to stop the presses), Taylor and his colleagues had detected one of the most iconic effects of general relativity, known as the “periastron advance” of the orbit.

  According to Newton’s gravitational theory, the orbit of a planet about its star is generally an ellipse, with the long axis of the ellipse always pointing in the same direction. For a binary star system, each body moves on an ellipse with the center of mass of the system as a focus (see Figure 5.1), but both ellipses are fixed in orientation. Any deviation from a pure Newtonian gravitational force between the two bodies, such as the gravitational tug of a third nearby body, or a modification of Newton’s laws provided by a theory like general relativity, can cause the ellipse to rotate or “precess,” as illustrated in Figure 5.4. As a result, the periastron, or point of closest approach, will not always be in a fixed direction, but will advance slowly with time.

  Figure 5.4 Top panel: Advance of the perihelion of Mercury’s orbit around the Sun. Because the Sun is so massive, it barely moves. The rate of advance has been greatly exaggerated. Bottom panel: Advance of the periastron of two stars A and B of comparable mass orbiting around their center of mass C.

  In the case of Mercury orbiting the Sun, astronomers had already established by the middle of the nineteenth century that its point of closest approach, called the perihelion, was advancing at a rate of 575 arcseconds per century. It was reasonable to assume that this was the result of the perturbing effects of the other planets in the solar system (Jupiter, Venus, Earth, etc.), and French astronomer Urbain Jean Joseph Le Verrier, who was the director of the Observatory of Paris, set out to calculate these effects. In 1859 he announced that there was a problem. The sum total of the effects of the planetary perturbations fell short of the observed advance of the perihelion by about 43 arcseconds per century (here we are quoting the modern value of the discrepancy). Le Verrier had recently become famous by predicting that some anomalies in the orbit of Uranus were being caused by a more distant planet, a prediction confirmed months later when German astronomers, using his calculations as a guide, discovered Neptune. It was therefore natural for Le Verrier and his contemporaries to postulate that the anomaly in Mercury’s orbit was due to a planet between Mercury and the Sun. They even gave the planet the name Vulcan, after the Roman god of fire. Despite numerous claimed sightings of Vulcan over the next 60 years, no credible evidence for the planet was ever found.

  Einstein was well aware of the problem of Mercury’s anomalous perihelion advance, and in fact he used it as a way to test and ultimately reject earlier versions of his theory, notably a “draft” theory he had devised in 1913 with mathematician Marcel Grossmann. In November 1915, when everything seemed to be falling into place theoretically for his latest attempt, the tipping point occurred when his calculations showed that the theory gave the correct value for the missing perihelion advance. He later wrote that this discovery gave him “palpitations of the heart.”

  In 1915, the agreement that Einstein found was fairly crude, because the observations of Mercury’s orbit were not very accurate. But since the 1970s Mercury’s perihelion advance has become another high-precision confirmation of general relativity, made possible by high-precision radar tracking of planets and spacecraft. The most recent test was made using Mercury MESSENGER. In 2011, MESSENGER became the first spacecraft to orbit Mercury, and radar measurements of the orbiter were made until the spacecraft ended its mission in 2015 with a controlled crash on the surface of Mer
cury. The data from that mission yielded a measurement of Mercury’s perihelion advance in agreement with general relativity to a few parts in 100,000. Improved measurements down to the level of parts per million may be possible with data from the joint European-Japanese BepiColombo mission to place two orbiters around Mercury, launched in late 2018.

  If Einstein’s theory indeed played a role in Hulse and Taylor’s system, then measuring the binary analogue of Mercury’s advance was of high priority, and during a two and a half month observing program that ended on 3 December 1974, Taylor and his colleagues tried to pin it down. Coming up was the seventh installment of the Texas Symposium on Relativistic Astrophysics that had begun in Dallas in 1963. After cycling twice through a trio of cities that included Austin and New York, it was back in Dallas. The data analysis was completed just in time for Taylor to reveal to the audience on 20 December that the rate of periastron advance for the binary pulsar was 4 degrees per year. This advance rate is about 36,000 times larger than the rate for Mercury, but that’s not a surprise. Because of the higher orbital speed and smaller separation in the binary system, the raw effects of general relativity are roughly 100 times larger than for Mercury, and the binary system completes 250 times more orbits per year, so the cumulative effect on the periastron builds up faster. Taylor would return to the Texas symposium four years later with an even more impressive announcement.

  While it was great to see Einstein’s theory in action in a new and exotic arena outside the solar system, Taylor’s measurement didn’t actually provide a real test of the theory. The problem is that the prediction of general relativity for the periastron advance for a binary system depends on the total mass of the two bodies; the larger the mass, the larger the effect. It also depends on other variables, such as the orbital period and the ellipticity of the orbit, but these are known from the observations. Unfortunately, we do not know the masses of the two bodies with any degree of accuracy. All we know is that they are probably comparable to that of the Sun in order to produce the observed orbital velocity, but there is enough ambiguity, particularly in the tilt of the orbit with respect to the plane of the sky, to make it impossible to pin the masses down any better from the Doppler shift measurements alone. Well, if we can’t test general relativity using the periastron advance measurement, what good is it?